Tagged Questions

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map. Riesz-Thorin gives us that ...
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Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$ by $\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.$ It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...
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Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0$ on $\mathbb{R}^{2n}$...
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Hilbert Transform and multiplier in $\mathbb{C}(X)$

I found myself trying to solve an equation of that kind : $$H f= R f,$$ where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...
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Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...
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Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
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How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
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Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...
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Fourier transform of tanh [closed]

tanh is not absolutely integrable, so a direct fourier transform does not exist. But even for the signum function, which is not absolutely intrgrable, we can get the fourier transform by applying some ...
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Square root of dirac delta function

Is there a measurable function $f:\mathbb{R}\to \mathbb{R}^+$ so that $f*f(x)=1$ for all $x\in \mathbb{R}$, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1$$ for all $x\in \mathbb{R}$.
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Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
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Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
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How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows: {X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...
Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries. Suppose now that, on top of having nonnegative entries, ...